In fact, assuming that the averaging kernel is the Green function of a differential operator, from the integral convolution it is possible to obtain a consequent differential formulation [1]}, [2]}.
For bounded structural domains, to the differential formulation above is needed to add a proper set of Constitutive Boundary Conditions (CBCs) [3]}.
Paradoxical results and unacceptable nonlocal responses are obtained [4]}, [5]}
if CBCs are ignored.
Several nonlocal theories have been adopted to overcome the aforementioned difficulties, such as: two-phase models [6]}, [7]}, strain and stress gradient theories
[8]}, [9]}, [10]}, [11]}, [12]},
nonlocal gradient techniques [13]}, [14]}, [15]},
strain-difference approaches [16]}, [17]}, [18]}, displacement-based nonlocal models [19]}, [20]}, [21]}, [22]}, stress-driven formulation of nonlocality [23]}, [24]}.
Advantageously, the stress-driven approach has been shown to be able to effectively model the nonlocal behaviour of small-scale structures and provides exact solutions for problems of applicative interest
in Nano-Engineering [25]}, [26]}.
The stress-driven nonlocal integral model is therefore adopted in this paper to significantly tackle size-dependent random vibrations in inflected elastic small-scale beams.
A first effort on the matter, disregarding random phenomena, was performed in [27]}.
